Langevin Analysis for Time-Nonlocal Brownian Motion with Algebraic Memories and Delay Interactions

Eur. Phys. J. B (2016) 89: 87 DOI: 10.1140/epjb/e2016-70079-5

Langevin analysis for time-nonlocal Brownian motion with algebraic memories and delay interactions

Matthew Chase, Tom J. McKetterick, Luca Giuggioli and V.M. Kenkre Eur. Phys. J. B (2016) 89: 87 DOI: 10.1140/epjb/e2016-70079-5 THE EUROPEAN PHYSICAL JOURNAL B Regular Article

Langevin analysis for time-nonlocal Brownian motion with algebraic memories and delay interactions

Matthew Chase1,a, Tom J. McKetterick2,3, Luca Giuggioli2,3,4, and V.M. Kenkre1 1 Consortium of the Americas for Interdisciplinary Science and the Department of Physics and Astronomy, University of New Mexico, Albuquerque − New Mexico 87131, USA 2 Bristol Centre for Complexity Sciences, University of Bristol, BS2 8BB, Bristol, UK 3 Department of Engineering Mathematics, University of Bristol, BS8 1UB, Bristol, UK 4 School of Biological Sciences, University of Bristol, BS8 1TQ, Bristol, UK

Received 2 February 2016 / Received in ﬁnal form 13 February 2016 Published online 6 April 2016 – c EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2016

Abstract. Starting from a Langevin equation with memory describing the attraction of a particle to a center, we investigate its transport and response properties corresponding to two special forms of the memory: one is algebraic, i.e., power-law, and the other involves a delay. We examine the properties of the Green function of the Langevin equation and encounter Mittag-Leffler and Lambert W-functions well-known in the literature. In the presence of white noise, we study two experimental situations, one involving the motional narrowing of spectral lines and the other the steady-state size of the particle under consideration. By comparing the results to counterparts for a simple exponential memory, we uncover instructive similarities and differences. Perhaps surprisingly, we find that the Balescu-Swenson theorem that states that non-Markoffian equations do not add anything new to the description of steady-state or equilibrium observables is violated for our system in that the saturation size of the particle in the steady- state depends on the memory function utilized. A natural generalization of the Smoluchowski equation for the time-local case is examined and found to satisfy the Balescu-Swenson theorem and describe accurately the first moment but not the second and higher moments. We also calculate two-time correlation functions for all three cases of the memory, and show how they differ from (tend to) their Markoffian counterparts at small (large) values of the difference between the two times.

1 Introduction The word “essentially” refers here to the fact that initial conditions dictate in this context an extra term not The time evolution of the probability density of a parti- present in the diffusion equation. The term decays at the cle attracted to a center via harmonic forces while being rate γ in a well-known way. simultaneously subjected to white Gaussian noise is of in- Our interest in the present paper is in systems in terest in a large variety of contexts. The label Ornstein- which the attraction to the center proceeds via a time- Uhlenbeck is attached to the system or process under nonlocal process. Memory-possessing Langevin equations such situations and the governing equation is said to be have come under investigation in various unrelated con- the Smoluchowski equation. It is ubiquitous in statistical texts in the past, but it is appropriate to say that mechanics [1,2] and, in a one-dimensional system takes modern work on the topic appears to have begun with the form Budini and Cáceres [3]. In a report with far-reaching conclusions, they studied generalized Langevin equations ∂P(x, t) ∂ ∂P(x, t) producing many interesting results, but restricted their = γxP(x, t)+D (1) ∂t ∂x ∂x investigations primarily to the exponential memory, fo- cusing on various kinds of non-Gaussian noise includ- where γ measures the rate of attraction to the fixed cen- ing radioactive, Poisson, and Abel noise. They touched ter and D is the diffusion constant. Its solutions are well- upon algebraic memory using fractional derivatives but known as being essentially identical to those of the sim- assumed zero dissipation in their analysis, i.e., γ =0. ple diffusion equation (no attractive center) provided the In a more recent study, they analyzed stationary proper- time t itself undergoes a saturation transformation via ties of Langevin equations with memories of the algebraic the Ornstein-Uhlenbeck prescription: t → (1 − e−2γt)/2γ. and delay type [4]asdidDrozdov[5] using functionals to characterize various noise distributions. There is also a e-mail: [email protected] an intriguing report in the literature by Fiscina et al. [6] Page 2 of 15 Eur. Phys. J. B (2016) 89: 87 of related ideas to the behavior of vibrated granular ma- one-variable description but to introduce a memory func- terial: they found that the observed asymptotic spectral tion relating the time derivative of x to an appropriate density could be well described using a Langevin equation function of x. Such a situation has been discussed in a with fractional derivatives. Work exploring the dynamic recent review of the mathematics of animal motion by behavior of a non-Markoffian Langevin equation was re- two of the present authors [14]. The origin of the memory ported by Bolivar [7] for arbitrary Gaussian noise corre- function or of time-nonlocality would be, in this case, the lation functions with a focus on the differentiability of incorporation of inertial terms. A quite different source of the displacement. A path integral analysis of a Langevin time-nonlocality is finiteness of the speed of propagation equation with exponential memory has also appeared [8]. of signals that are related to the attraction process. Ex- Fractional derivatives have been used [9] to introduce what amples may be found in delay formalisms as in a study are in essence memory effects into fractional probability of Alzheimer walks [15] and a recent analysis of pairwise density equations. General expositions of the subject that movement coordination [16] applicable, e.g., to a system have been highly useful for a number of years are available of foraging bats [17]. in various articles [10–13]. Although we will not use Fokker-Planck equations As is well-known, stochastic analysis may be under- for our analysis, we will begin our considerations with taken either via Langevin equations or via what has often the Smoluchowski equation (1). We do this because it been called a formalism based on Fokker-Planck equa- is easy to introduce our task in that manner and also tions. The former are ordinary differential equations for because there has been a lot of recent activity on that stochastic variables. The latter are partial differential equation. Thus, an analysis has shown [18]howthe equations for the deterministic probability density of those Smoluchowski equation may be applied to trapping situa- variables, an average of the stochastic variables carried out tions and a recent application to the spread of epidemics with the help of the probability density leading to expec- has been made [19] leading to an interesting description tation values. In the present paper, except for a discussion of the transmission of infection in diseases such as the leading into the problem as well as an interesting obser- Hantavirus [20]. vation at the end of the analysis, we will not use Fokker- Planck equations. A number of subtleties and controver- The rest of the paper is laid out as follows. We first sies appear in the discussion of Fokker-Planck equations show in the next section why an intuitively natural mem- and we plan to address them in a separate publication. ory generalization of an equation such as equation (1) Our focus in the present paper will be entirely on the does not work. We place our full focus therefore on the Langevin approach: the ordinary differential equation for Langevin equation with memory. We are generally inter- the basic stochastic variable under consideration is solved ested in biological processes far from thermodynamic equi- explicitly for the given memory and noise, and the desired librium. The near-equilibrium requirement of fluctuation- function of the variable, for instance an arbitrary power, dissipation relations between the noise and the memory is averaged taking into account the properties of the noise. then need not apply. To facilitate calculations we take the Thus, our general interest is in treating systems in noise to be white. Although this restriction is not essential, we limit ourselves to this case on one hand because of which the Langevin equation for the coordinate xi of the ith particle is its general usefulness in providing insights into a variety of biological systems [21] ranging from physiological [22,23] t to ecological scales [24], and on the other hand because of dxi(t) = −γ dt φ(t − t )xi(t )+ξi(t), (2) the already rich scenario that we uncover even when the dt 0 noise is not colored. In Sections 3 and 4 respectively, we where ξi(t) denotes the noise and φ(t) is the memory func- obtain explicit usable prescriptions, specifically for alge- tion. The case when the memory is a δ-function, i.e., when braic and delay-type memories in the Langevin equation. the Langevin equation is time-local, corresponds to equa- We find the Green function for the case of an algebraic tion (1). Our specific interest is in two forms of the mem- memory to be the Mittag-Leffler function. In the case of ory: algebraic, i.e., power law, and incorporating a delay, a single delayed delta function, it is associated with the as we will explain below. Lambert function. A regime in which there is a mono- One example of how non-local (in time) Langevin tonic decrease on the one hand and decaying oscillations equations may come about in physical processes is pro- on the other is found in each of the two Green functions. vided by a scenario in which one refrains from making This is precisely the behavior shown by the simple and the high-damping approximation in the Langevin equa- well understood case, for instance for a damped harmonic tion. Normally, as a result of Newton’s law, the latter is oscillator, when the memory is an exponential. a second order equation for the time dependence of the We discuss in Section 5, two applications of our results particle coordinate x. If the damping is very strong, the in the context of two specific experiments that could be normal practice is to reduce it to a first order equation performed on our system in principle. The first is about by neglecting the inertial term. If the inertial term is kept motional narrowing in the frequency-dependent suscepti- intact, both variables, the coordinate x and the velocity v, bility of our system wherein the particle is charged and a are typically treated on an equal footing. An alternative, time-periodic electric field is applied. The second queries viable when observables dependent only on the coordi- the steady-state size of the system as measured by the nate (and not the velocity) are of interest, is to stick to a mean square displacement of the particle around its fixed Eur. Phys. J. B (2016) 89: 87 Page 3 of 15 center under the combined action of the time-nonlocal at- Needless to say, this is the high-friction (sometimes re- traction and the diffusion. Comparison of the results for ferred to as the Aristotelian) limit corresponding to the the three memories considered, the algebraic, the single time-local approximation to the Langevin equation. If delay, and the simple exponential, brings out interesting standard procedures [27] are now used to derive the dif- similarities and differences. In the discussion which consti- fusion term from the noise in the Langevin equation, one tutes Section 6, we analyze a natural but incorrect general- obtains (1). This description may be found for instance in ization of the Smoluchowski equation yet find the remark- the textbook by van Kampen [28]. able result that its predictions are correct in the context However, if the Langevin equation has memory effects of the first of the two envisaged experiments. as in equation (2), xi(t) cannot be replaced by x in the Contexts in which the analysis presented in this paper t-nonlocal velocity equation (2). The connection is to val- should find applicability are, generally speaking, biologi- ues of x occupied by the ith particle at all times in the cal systems far from equilibrium. These include movement past. It is then impossible to replace xi(t ) for all t by x: of animals having a preference to places visited in the the delta-functions in x do not allow xi(t) to be replaced past [25] and various versions of what has been called in by x. This technical failure to arrive at non-Markoffian the mathematical literature ‘self-reinforced random walks’ field equations is a direct consequence, inevitable at least as are met in Alzheimer-related investigations [26]. The at this level, of the memory nature of the interaction. complexity of the biological nature of the systems involved A viable route to the description of memory effects lies, necessitates a memory description at the Langevin level however, in refraining from a generalization of equation (1) and the fact that we are not necessarily near equilib- and in restricting one’s attention entirely to the Langevin rium suggests that a fluctuation-dissipation relationship approach. There are in the literature a number of discus- between the memory and the noise need not apply mak- sions centered on probability density evolution [29–32]. ing our simplification of Gaussian noise a useful first step For instance, for the specific case of Gaussian white in the analysis. noise, a Fokker-Planck like equation of the Smoluchowski form results. San Miguel and Sancho [32] address a system consisting of a Brownian harmonic oscillator with fi- 2 Viable route for the description of memory nite inertia. Involved is a second-order linear Langevin effects equation subject to an additive Gaussian stochastic forc- ing. Although their derivation is specifically performed The temptation to generalize the Smoluchowski equa- for the Brownian harmonic oscillator and results in an tion (1), to incorporate memory effects present in its exponentially decaying memory kernel, Langevin equation as in (2), by making (1) itself nonlocal −bt in time is natural but meets with the following problem. φ(t)=be , (5) Let us, in addition to the coordinate xi(t), define the ve- it can be easily generalized to any arbitrary linear memory locity of the ith particle at time t as vi(t)=dxi(t)/dt,the kernel. The b in equation (5)obeysγb = ω2, the square of latter being given as a stated function of xi(t). The obvious manner of transforming from a particle description to the oscillator frequency. a field description in the space of the field variable x is to Let us avoid the probability density treatments and begin with the microscopic definitions of the (probability) start from equation (2). Since there are no inter-particle density P (x, t) and the current density j(x, t), interactions, we consider a single particle and drop the label i without loss of generality. Let λ(t) be the Green function of the homogenous (without noise) part of equa- P (x, t)= δ(x − xi(t)), i tion (2). An immediate consequence of equation (2)is dxi(t) 1 j(x, t)= δ(x − xi(t)), dt λ( )= (6) i + γφ( ) and to obtain the continuity equation as a consequence: in the Laplace domain: tildes denote Laplace transforms and is the Laplace variable. This result leads to the ∂P(x, t) ∂j(x, t) + =0. (3) solution of equation (2) in the time domain as ∂t ∂x t One then expresses the x-derivative of j(x, t) by replacing xi(t), wherever it occurs, by x given that δ(x − xi(t)) x(t)=λ(t)x(0) + dt λ(t − t )ξ(t ). (7) is present as a multiplying factor. This allows us to com- 0 bine the continuity equation with the specific relation between vi(t)andxi(t) that forms the constitutive relation, From here onwards, we only consider systems in which the and thereby to obtain the Smoluchowski equation. Specifi- noise ξ(t) has zero mean, ξ(t) = 0, and is white, which cally, for the simple time-local case in the absence of noise, means that ξ(t)ξ(s) =2Dδ(t − s) where the constant D describes the strength of the noise. Expectation values of dxi(t) arbitrary powers of x at a specified time can be calculated vi(t) ≡ = −γxi(t). (4) dt explicitly, making the reasonable additional assumption Page 4 of 15 Eur. Phys. J. B (2016) 89: 87

∞ that the noise is uncorrelated also with the initial value of strength 0 dtφ(t). We consider such memories as our of the observable. The result for the average displacement starting point for the present section: and average square of the displacement is: α(αt)ν−1 φ(t; ν)= . (12) x(t) = x0λ(t), (8) Γ (ν) t 2 2 2 2 Here α is a positive constant with units of inverse time Δx (t) = x(t) −x(t) =2D dsλ (s), (9) and Γ (ν) provides the appropriate normalization. We 0 analyze the Green function λ(t). The Laplace transform1 of equation (5), and inser- where we take the (localized) initial condition as x(0) = tion into equation (6), gives the Laplace domain Green x0. In light of the fact that equation (2) already has t =0 function as, as a special instant at which the memory is initialized, we observe that there are now two, generally different, times 0 ν 1 1 and t0, the latter being the time at which x(t0) is first λ( ; ν)= γαν = ν+1 ν = γαν , (13) + ν + γα 1+ ν+1 measured, e.g., the initial observation time. For the sake of simplicity, we have used t0 = 0; in general the two times which is the Laplace-domain representation of the Mittag- may well be different. For non-Markoffian processes, the Leffler function of one parameter, written in usual nota- ν ν+1 general case for which the two times are different leads tion as Eν+1(−γα t )[33]. In the time domain, setting to interesting subtleties which we do not analyze in the γαν ≡ ζν+1, this results in the series, present work. In the second line we have displayed the n difference of the average of the square of the displacement (1+ν) ∞ − (ζt) and the square of its average. We represent it by the sym- 2 2 λ(t; ν)= . (14) bol Δx and refer to it, rather than to x ,asthemean n=0 Γ (n(1 + ν)+1) square displacement (MSD). Expectation values of two- time quantities such as the correlation function x(t)x(s) This expression is derived through a binomial expansion of can also be obtained straightforwardly: the denominator in equation (13). One obtains a formal series in increasing powers of (ζ/ )1+ν . A term-by-term s 2 inverse Laplace transform of this formal series results in x(t)x(s) = x0λ(t)λ(s)+2D dt λ(t − t )λ(s − t ), equation (14) for all ν>−1. The resulting series converges 0 for all finite times. (10) For the parameter range of interest, the Green func- ≡ − tion we have calculated shows three interesting types of f(t, s) x(t)x(s) x(t) x(s) ∈ − s behavior. The first is an overdamped decay, ν ( 1, 0), ∈ the second is underdamped oscillations, ν (0, 1), and the =2D dt λ(t − t )λ(s − t ). (11) third unstable oscillations, ν ∈ (1, ∞). We depict λ(t)for 0 the cases of overdamped decay (left) and underdamped oscillations (right) in Figure 1 over 16 dimensionless time The above results do not require that the noise be units ζt. The overdamped regime exhibits sharper initial Gaussian. If, however, it is known to be Gaussian we can decays, but longer tails, as the value of ν is made more also write, for arbitrary powers of the displacement, negative. Both, the amplitude of oscillation and the time ⎧ for which they persist, increase for increasing ν in the un- p (2p)! 2m 2m ⎪ m=0 (2m)!(p−m)! x0 λ (t) derdamped regime. The value of dλ(t)/dt at t = 0 changes ⎪ p−m ⎪ t discontinuously when ν approaches 0 from either direc- ⎨ × D dsλ2(t) even n p ≡ n , n 0 2 tion. For positive values of ν it vanishes. For negative val- x (t) = p (2p+1)! ⎪ x2m+1 λ2m+1(t) ues it tends to infinity. This behavior is sharply different ⎪ m=0 (2m+1)!(p−m)! 0 ⎪ t p−m from that in the case of the simple exponential memory ⎩ × 2 ≡ n−1 D 0 dsλ (t) odd n p 2 , characteristic of the damped harmonic oscillator. For the latter, dλ(t)/dt at t = 0 always vanishes. Not shown are and thereby solve the entire problem on the basis of the the unstable oscillations for values of ν>1. Three spe- Gaussian property. cial values exist. For ν = −1, we have standard Brownian diffusion, i.e., a Wiener process or an unconfined random walk. For ν = 0, we have the standard Smoluchowski equa- 3 Algebraic memories in the Langevin tion, the Ornstein-Uhlenbeck process. For ν =1, we have equation pure oscillations with no damping. 1 The Laplace transform of (12)onlyexistsforν>0. How- The family of algebraic functions provides a useful case ever, the form of the Laplace-domain noiseless Green function, study of the class of memories that cannot be approxi- (13), suggests extending the domain of validity to ν ≥−1. In mated via a Markoffian procedure. The latter means the this range, the Green function is unity when t =0(exceptfor replacement of φ(t) for long times by a delta-function in t ν = −1whereλ(t)=1/2). Eur. Phys. J. B (2016) 89: 87 Page 5 of 15

∞ ν sin νπ −rζt r λ(t; ν)=− dre π r2(ν+1) − 2rν+1 cos νπ +1 0 ⎧ ⎨⎪0 −1 <ν≤ 0 −ζt cos νπ 2 e ν+1 ζt νπ <ν≤ + ν+1 cos( sin ν+1 )0 2 (15) ⎩⎪ −ζt cos νπ −ζt cos 3νπ 2 e ν+1 ζt νπ e ν+1 ζt 3νπ <ν≤ ν+1 cos( sin ν+1 )+ cos( sin ν+1 ) 2 4

Overdamped Regime Underdamped Regime 1 1 = -0.8 0.8 = 0.2 = -0.6 = 0.4 0.8 = -0.4 0.6 = 0.6 = -0.2 =0.8 0.4 0.6 0.2 (t) (t) 0 0.4 -0.2 0.2 -0.4 -0.6 0 -0.8 0 5 10 15 0 5 10 15 t t Fig. 1. The noiseless Green function, λ(t), for varying ν in equation (14) over the overdamped (left) and underdamped (right) regimes in the range [−0.8, −0.2] and [0.2, 0.8] in steps of 0.2. Time is plotted in units of 1/ζ.Asν is made more negative, λ(t) approaches 0 more slowly. As ν is made more positive, the amplitude of the oscillations increases and they last longer.

A second representation of λ(t) is found by explic- The integral in equation (15) is not reducible in terms itly closing the Bromwich contour. For all non-integer of known functions for arbitrary ν. For the particular case values of ν,(13) has at least two singularities of inter- of ν = m +1/2, where m is an integer, a transform of est: branch points at zero and infinity, connected by a u = r1/2 simplifies the integrand to branch cut chosen to be along the negative real axis. This ∞ 2(m+1) choice of branch cut is made to restrict the domain of to − m+1 2 −u2ζt u | | ( 1) due 2(2m+3) . the Riemann sheet with arg( ) <π. Additionally, sim- π 0 u +1 ple poles exist for all relevant values of ν at the points ln( )=±i(1+2m)π/(ν +1)where m is an integer greater The denominator of this integral is easily factorable into than zero. However, not all fall on the relevant Riemann the (2m + 3)th roots of 1. This results in the standard sheet. When ν<0, there are no additional simple poles. integral representation of the Faddeeva function, w(iz)= As ν passes through each successive even integer, two ad- erfcx(z), which corresponds to the scaled error functions ditional poles move on to the relevant Riemann sheet. with complex arguments [34]. We give here the ν = −1/2 Therefore, the Bromwichintegralofequation(13)canbe (m = −1) and ν =1/2(m =0)cases: performed to quadrature and results in 1 ζt 1 λ t; − = e erfc (ζt) 2 , (16) see equation (15) above 2 1 1 − ζt 3 2 ζt 1 where λ(0; ν)equals1forallν. The integral in equa- λ t; = e 2 cos + w i(ζt) 2 tion (15) is the Laplace transform of a positive definite 2 2 function and, for non-integer values of ν, is therefore 1 1 i2π 2 − i2π 2 non-negative at all times. As mentioned previously, addi- − w i ζte 3 − w i ζte 3 . tional exponential terms become relevant as ν is increased (17) further. We have given two separate representations of λ(t)in The long-time behavior of the Mittag-Leffler function, thetimedomain:theseries,(14), and the integral expres- valid for non-integer values of ν in the region (−1, 1), is sion, (15). These are compared in Figure 2,overarange well-known [33], of approximately 30 dimensionless time units for 2 values ± ± p n of ν in each regime, 0.1 and 0.9. The two representa- 1 −1 tions match up well over shorter time periods. However, at λ (t →∞; ν)=− Γ [1 − n(ν +1)] (ζt)(ν+1) longer times, the numerical implementation of the series n=1 leads to divergent results as a consequence of round-off + O t−p(ν+1) . (18) errors. Page 6 of 15 Eur. Phys. J. B (2016) 89: 87

Overdamped Regime Underdamped Regime 1.2 1.2 Integral 1 Integral 1 Series Series 0.8 0.8 0.6 0.6 0.4

(t) 0.4 (t) 0.2

0.2 0 -0.2 0 -0.4 -0.2 -0.6 -0.4 -0.8 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 t t

Fig. 2. Depicts the integral representation and the series representation of the Green function, λ(t), over short times for the overdamped regime (left) and the underdamped regime (right) for ν = ±0.2, ±0.8. The series representation is indicated by dots and the integral representation by solid lines. At approximately 30 dimensionless time units ζt, the series begins to diverge as a result of numerical round-oﬀ results.

=0.1 =0.5 10-6 =0.9 0 0.04 -2.5 0.02 -3 -0.01 0 -3.5

(t) -0.02 (t) -0.02 (t) -4 -0.04 -4.5 -0.03 -0.06 -5 -0.04 -0.08 -5.5 5 101520 5 1015202530 180 200 220 240 t t t =-0.1 =-0.5 =-0.9 0.2 0.4 0.41 0.35 0.405 0.15 0.4 0.3 0.395 (t) (t) 0.1 (t) 0.25 0.39 0.2 0.05 0.385 0.15 0.38 0 0.1 0.375 5 101520 5 101520 30 40 50 60 70 t t t Fig. 3. Comparison at long times of the integral expression, equation (15), of the noiseless Green function (solid line) with its series approximation (dashed line), equation (18). The latter consists of the ﬁrst 5 terms for 3 values of ν, 0.1 (left), 0.5 (center), and 0.9 (right) over diﬀering time ranges.

We see that an algebraic memory results in an algebraic Thus, the algebraic memory leads to a noiseless Green time-dependence of λ(t) at long times. The dominant term function that corresponds exactly to the Mittag-Leﬄer 1+ν in the series, proportional to 1/ (ζt) , leads to a decay function. Three separate regimes emerge: overdamped de- which is stronger when ν is larger. In the underdamped cay, −1 <ν≤ 0, underdamped oscillations, 0 <ν<1, regime, the leading term of (18) is negative. Therefore, and unstable oscillations, ν ≥ 1. at long times, λ(t) approaches zero from below, conﬁrm- ing the existence of at least one minimum. This term is positive for the overdamped regime. The correspondence 4 Memories that represent delay processes between the long-time approximation, equation (18)with 5 terms, and the full propagator is depicted in Figure 3 for We now focus on the particular case of a memory which se- 6valuesofν: ±0.1, ±0.5, ±0.9. The long-time approxima- lects only one time τ in the past via a Dirac-delta centered tion does not lead to oscillations, rather to an overall de- at τ.Inotherwords, cay towards 0. For smaller values of |ν|, the approximation becomes valid at earlier times. φ(t)=δ(t − τ). (19) Eur. Phys. J. B (2016) 89: 87 Page 7 of 15

To avoid dealing with a piece-wise process, i.e. Wiener the characteristic equation provide information about the dynamics for 0 ≤ t<τ, and delayed dynamics for t ≥ τ, evolution and stability of the system at long times. It we extend the range of integration in the Langevin equa- is known [35] that the system decays monotonically for tion (2), back in time to −τ<0 and consider, 0 ≤ γτ < e−1, undergoes oscillatory decay for e−1 ≤ γτ < π/2, and performs unstable oscillations for γτ > π/2. At t dx(t) long times the eigenvalue with the largest real part, cor- = −γ ds φ(t − s)x(s)+ξ(t), (20) dt −τ responding to the principal branch of the Lambert function W0, dominates the behavior of the system. In the such that at time t = 0 the evolution of x is now de- monotonic regime (γτ < e−1), this principal branch can pendent on the history of x between t = −τ and t =0. actually be evaluated analytically [40]viaW0(−γτ)= ∞ n−1 n After inserting φ(t)=δ(t − τ)inequation(20) one recog- − n=1 n (γτ) /n!. nizes that the system is governed by the stochastic delay Whilst the expression for λ(t)inequation(24)isex- Langevin equation, act and provides insights into the long time behavior of the system, it is of limited use for studying the dynam- dx(t) = −γx(t − τ)+ξ(t),t>0 ics of the system at shorter times because of the large dt number of eigenvalues required. An alternative expression x(t)=Φ(t), − τ ≤ t<0, (21) can be derived for λ(t) by expanding (23)asapowerseries and performing the Laplace inversion to give the following where Φ(t) is a given deterministic history function. To expression [15,16,37,38], solve this stochastic delay differential equation we use the ∞ Laplace transform and obtain [35] (−γ)k λ(t)= (t − kτ)kΘ(t − kτ), (26) t k=0 k! x(t)=x(0)λ(t)+ ds λ(t − s)ξ(s) 0 where Θ represents the Heaviside step function. 0 The alternative exact expression (26) requires only a − − − γ ds λ(t s τ)Φ(s), (22) finite number of terms for any finite time t with a func- −τ tional dependence given by a polynomial of degree k in which is formally equivalent to equation (7), with the addi- each interval kτ ≤ t ≤ (k +1)τ. Expansion of the sum tional term representing the history-dependence. In equa- in equation (26) shows in fact explicitly that λ(t)=1 for t ∈ [0,τ], and = 1 − γ(t − τ)fort ∈ [τ,2τ], and tion (22) the Laplace transform of the Green function λ(t) 2 2 is given by, =1− γ(t − τ)+α (t − 2τ) /2fort ∈ [2τ,3τ], and so on. From these polynomials we can study properties of 1 the Green function, such as the time at which it first λ˜( )= . (23) + γe−τ reaches zero, denoted t0, when the system is in the oscillatory regime. The Green function is constant on the We now turn to the analysis of single delay processes. first interval and thus clearly t0 >τ. The Green function The analytic calculation of λ(t)fromequation(23)has on the second interval, when τ

Our interest lies in measuring the frequency-dependent 1 susceptibility which is the ratio of the Fourier transforms of the polarization and the applied electric ﬁeld. The term 0.5 E(t) is essentially the electric ﬁeld and absorbs unimpor- )

t tant proportionality constants. The spectral line at fre- ( 0

λ xˆ(f) quency f is proportional to Eˆ(f) where the circumflexes 0.5 γτ =0.3 denote Fourier transforms and f is the frequency. We use γτ =0.6 γτ =1 f rather than the more usual ω to distinguish it from 1 γτ =1.6 the oscillator frequency that we have already used in our treatment. 0 2 4 6 8 10 t/τ A well-known phenomenon known in systems with a simple exponential memory, as in the case of a damped Fig. 4. The noiseless Green function λ(t) for the single de- harmonic oscillator, is motional narrowing: spectral lines, lay process. Four choices of the parameter γτ are considered, representing the stable non-oscillatory, stable oscillatory and sharp if the damping in the system is vanishing or small, unstable oscillatory regimes. The first zero crossings of the os- broaden as the damping is increased but, after a critical cillatory curves occur in the second interval for γτ ≥ 1, specif- value of the damping is exceeded, separate lines coalesce ically at t0/τ =1.625 and 2 for γτ =1.6and1.0, respectively. and increased damping narrows the line. The question we In contrast, the oscillatory curve with γτ =0.6 first crosses ask here is whether this motional narrowing occurs also for the zero in the third interval. our algebraic and delay cases. One sees from equation (28) that what is required is the one-sided Fourier transform of the Green function λ(t): the frequency-dependent suscep- xˆ(f) our predictions for the three cases of algebraic, single de- tibility is proportional to .Fromequation(27)for lay, and exponential memory. The latter, given by equa- Eˆ(f) tion (5), has appeared not only in the damped harmonic the exponential memory and equations (14)and(26)for oscillator treated by the authors of reference [32], but in the algebraic and single delay memories, respectively, we numerous contexts, some to describe transport with an ar- have: bitrary degree of quantum mechanical coherence [41–43]. λˆ(f)= Although its introduction in the latter references has oc- ⎧ √ curred in the context of site-to-site motion of a quasi- ⎪ f 2+b2 ⎪ √ Exponential, (29a) particle, rather than of attraction towards a center as in ⎨ (f 2−ω2)2+f 2b2 the present paper, many expressions obtained earlier, and √ 1 Single Delay, (29b) ⎪ f 2−2fτμ2 sin fτ+τ 2μ4 much of the intuition, can be ported over here. The Green ⎩⎪ √ (|f|)ν 2(ν+1) ν+1 πν 2(ν+1) Algebraic, (29c) function λ(t) is well-known to be given by the simple |f| −2(|f|ζ) sin 2 +ζ expression where we have given the absolute value of the Fourier −bt/2 λ(t)=e [cos Ωt +(b/2Ω)sinΩt] , (27) transforms. In equation (29b) we have introduced the coherence parameter μ for the delay case. It is analogous where Ω = ω2 − b2/4. Standard features well-known to ω in the exponential case and ζin the algebraic case from other fields of study, ranging from quantum quasi- and is specifically defined via μ = γ/τ. particle transport to RLC circuits in electrical networks, We show the results of equation (29) in Figure 5 for are easily noticed in equation (27): the original undamped the exponential (left), single delay (center) and algebraic oscillator frequency ω, its reduction to Ω when damping (right) memories. Units along the vertical axis are arbi- is introduced via the damping exponent b/2, the passage trary. The left panel depicts motional narrowing for the from the damped oscillatory regime to the overdamped exponential memory: two peaks broaden and then coa- regime when b/2 >ωwhen the trigonometric functions lesce into a single peak as the damping is increased. Sim- change into their hyperbolic counterparts leading to fa- ilar transitions are seen for both the single delay memory miliar phenomena such as motional narrowing of spectral (coherence measured by the value μτ) and the algebraic lines. memory (coherence measured by the value ν). The single delay process is quite similar with the exception that 5.1 Motional narrowing of spectral lines it develops additional symmetric peaks, indicated by the arrows. The algebraic process, while also similar in the We envisage two experiments that could be performed, in overall aspects, exhibits sharp differences for small values principle, on our system. In one, our particles are charged of f. The source of this peculiar behavior is the fact that (but noninteracting among themselves), we apply a time- the integral of λ(t) over all time changes drastically as one varying electric field and measure the polarization and crosses from the oscillatory to the monotonic region. The therefore the susceptibility. Thus, equation (2), with the integral is 0 for positive ν and infinite for negative values label i suppressed, is augmented by an appropriate term of ν. to give The motional narrowing phenomenon [44]wehave dx(t) t discussed above is ubiquitous and appears in mag- = −γ dt φ(t − t)x(t)+E(t)+ξ(t). (28) dt 0 netic resonance observations [45], neutron scattering Eur. Phys. J. B (2016) 89: 87 Page 9 of 15

Exponential Single Delay Algebraic 12 12 8 /b = 8 = 2 = 0.9 /b = 2 = 1.2 7 = 0.5 10 /b = 1/2 10 = 0.75 = 0.1 /b = 1/12 = 0.25 6 = -0.3 8 8 5

6 6 4

3 4 4 2 2 2 1

0 0 0 -2 -1 0 1 2 -5 0 5 -2 -1 0 1 2 f/ f/ f/ Fig. 5. Motional narrowing in the dependence of the a.c. susceptibility on the frequency, f, of the applied electric field. All three cases, exponential (left), single delay (center), and algebraic (right), display two peaks in the coherent limit which, as damping is increased, initially broaden more and more and move towards each other (to the center, i.e., the region f =0).On increasing the damping beyond a critical value in each case, however, the line narrows rather than broadens as the damping is increased. Frequency is plotted on the horizontal axis in units of the coherent parameter ω, μ and ζ for the three respective cases. Units along the vertical axis are arbitrary. Two arrows in the central panel locate the additional peaks that develop for the single delay process. The sharp transition at f = 0 for the algebraic memory, which is a jump from a vanishing to an infinite value, can be seen when comparing the ν = −0.3casewiththeν =0.1case. experiments [46–51], and in numerous other contexts such Although any two of the three parameters ω, b and Ω as optical absorption whenever there is underlying dynam- uniquely determine the third, we have used all three here ics of a system undergoing spectral diffusion. Such spec- and elsewhere to avoid cumbersome square roots in the tral diffusion can arise not only from thermal motion in display. In the overdamped limit, i.e., when b>2ω,the an inhomogeneous medium as has been sometimes men- above trigonometric functions turn into hyperbolic func- tioned in the past but from a variety of sources including tions as Ω becomes imaginary. changes in the bath fluctuation rate [52]. The simple ex- The MSD for the case of the single delay is: ponential memory case we have mentioned for the sake of comparison above can be seen to arise explicitly in the k−1 (l+1)τ t magnetic resonance context. This can be noted clearly 2 Δx (t)=2D Θ(k) ds gl(γs)+ ds gk(γs) , in Appendix F of the text Principles of Magnetic Reso- lτ kτ nances by Slichter [45] where the exponential nature of the l=0 memory emerges simply on elimination of separate quanti- (31) where gk(t) is defined as: ties M±, summing them to get the total magnetization M. The original theory of motional narrowing was developed by Kubo [44], a number of cases for neutron scattering k k (m+n) (−1) m n were treated for simple memories by other authors such gk(γt)= (γt − mγτ) (γt − nγτ) , as Brown and Kenkre [47–51] and a recent lineshape the- m =0n =0 m!n! ory was given by Jung et al. [52] directed at the important observation of power-law statistics in spectral diffusion. in any interval, kτ ≤ t ≤ (k +1)τ. The MSD for the algebraic case is given by, 5.2 Spatial extent in the steady state

What is the spatial extent of the particle in the steady- see equations (32a) and (32b) next page state when the diﬀusive tendency to enhance it and the attraction to the center to reduce it have balanced themselves? A number of experimental techniques could be de- with vised in principle to measure the spatial extent or size. ν The size in the steady-state is given by the saturation 2 r sin νπ 2 A = ,C(r)= 2(ν+1) − (ν+1) , value of the MSD Δx . The time-dependent MSD for ν +1 r 2r cos νπ +1 the exponential memory is given by, νπ νπ β =cos ,κ=sin . D ω b be−bt ω4 4Ω2 − ω2 ν +1 ν +1 Δx2 (t)a = + − + ω b ω ω b2Ω2 4Ω2 Our interest lies in the steady-state size of the particle 3ω2 − 4Ω2 × cos 2Ωt + sin 2Ωt . (30) given by these expressions in the limit t →∞. Calling the 2bΩ saturation value of the MSD as the particle size S (in units Page 10 of 15 Eur. Phys. J. B (2016) 89: 87

⎧ ⎪ ∞ dr 2 ⎪ e−rζsC r − <ν≤ t ⎨ ( ) 1 0, (32a) 2 0 π Δx (t)=2D ds ⎪ ∞ 2 0 ⎪ −βζs dr −rζs ⎩ Ae cos κζs − e C(r) 0 <ν≤ 2, (32b) 0 π

Exponential Single Delay Algebraic 101 102 101 (s) (s) (s) 2 2 101 2 ds ds ds 0 0 0

100 100 100

024 0 0.5 1 -0.5 0 0.5 1 /b Fig. 6. Spatial extent of the particle in the steady-state, measured by the saturation value of the MSD at long times for: the exponential memory (left), the single delay memory (center) and the algebraic memory (right). On the vertical axis, the size is normalized to the steady-state size for the time-local case (φ(t)=δ(t)) with γ equal to the respective coherent parameter: ω for the exponential memory, μ for the single delay memory and ζ for the algebraic memory. Dotted lines indicate the location of the transition from monotonic (overdamped) to oscillatory (underdamped) regimes. of area for our 1-dimensional system), we have dependence on ω and b,withaminimumforb = ω.Forthe 2 2 ⎧ single delay, only in the stable regime, μ τ <π/2, is the ⎪ D ω b expression valid and, as expected, the MSD diverges as ⎪ + , (33a) ⎪ ω b ω one approaches the unstable limit. Both for the exponen- ⎪ ⎪ 2 2 tial and the single delay memories, the saturation value of ⎪ D 1+sinμ τ ⎪ , (33b) the MSD diverges as ω/b and μτ, respectively, approach ⎪ 2 2 ⎪ μ μτ cos μ τ zero. In both these cases, the tendency to conﬁnement ⎨⎪ ∞ ∞ D drdq 2C(r)C(g) around the attractive center disappears since γ vanishes. S = , (33c) ⎪ ζ π2 r + q At long times, the MSD for the algebraic case di- ⎪ 0 0 ⎪ 2 ∞ verges when ν →−0.5 from the right and when ν → 1 ⎪ D A 1 dr C(r)(β + r) ⎪ β + − 4A from the left. The divergence at the lower limit occurs ⎪ ζ 2 β π r2 +2βr +1 ⎪ 0 due to the long-time algebraic dependence of the Green ⎪ ∞ ∞ ⎩⎪ drdq 2C(r)C(q) functions, equation (18). The value of ν for which the + 2 , (33d) 0 0 π r + q steady-state size is minimum depends on the ratio of time constants, γ/α. When this ratio is equal to 1, the min- where (33c)isfor−0.5 <ν≤ 0and(33d)isfor0<ν≤ 1. imum is at ν = 0. This is obvious in Figure 6. For an For the parameter range ν ≤−0.5, the MSD diverges as, arbitrary γ/α,thevalueofν at the minimum is given by the transcendental equation, ln tν= −0.5, 2 ∝ lim Δx (t) 2|ν|−1 γ 2 d t→∞ t −1 ≤ ν<−0.5. ln = −(νmin +1) ln F (νmin), (34) α dν The MSD for the single delay memory, equation (33b), is where F (ν) is the functional form of the curve plotted in known to have the analytic expression shown above. It is the right panel of Figure 6.Anincreaseintheratioofγ/α obtained by solving [38] the diﬀerential equation governing shifts the minimum rightwards, a decrease shifts it to the the evolution of the covariance at long times. left. Plots of the long-time expressions in equation (33)are A fruitful comparison of the MSD dynamics can be ∞ 2 shown in Figure 6 for the exponential (left), single de- done by setting the parameters such that, 0 ds λd(s)= lay (center) and algebraic (right) memories, respectively. V ,where is the appropriate coherent parameter, is iden- The MSD for all three memories are normalized using the tical for all three cases. The saturation value of the three respective coherent parameters: ω, μ and ζ.Inallthree memories are not comparable over the entire parameter cases, an increase in D leads to a monotonic increase in the space. The single-delay process has a minimum MSD sat- MSD. The exponential memory process has a symmetric uration when 2μ2τ 2 =cos(μ2τ 2), i.e., a value V =1.19, Eur. Phys. J. B (2016) 89: 87 Page 11 of 15

3.5 ν = -0.4, 0.8, 0.9 center, of a particle via a memory function correspond- γτ = 0.19, 1, 1.3 b/γ = 2.4, 0.32, 0.15 ing to a time-nonlocal process. The case when the mem- 3 ory function, denoted by the symbol φ in the paper, is a simple delta function in time (time-local case), is text- 2.5 book material and leads to solutions of the correspond- ) s ( 2 ing Smoluchowski equation via the Ornstein-Uhlenbeck 2 dsλ t 0 time transformation. We have investigated here the case of two time-nonlocal memories of interest to certain current 1.5 applications in biological systems: the algebraic memory, 0 10 equation (12), and a delay case, equation (19), and com- 1 pared them with a standard exponential memory, equa- 0 1 2 10 10 10 0.5 tion (5), whose consequences are typically known in the lit- 5 10 15 20 25 30 t erature. We find similarities across the three cases as well Fig. 7. Comparing the dynamics of the MSD for the algebraic as some distinguishing characteristics of each. The connec- (straight line), exponential memory (dot-dashed line) and sin- tion to the time-local case represented by φ(t)=δ(t)is gle delay (dashed line) memories when their MSD have a mu- trivial in the case of the delay and the exponential cases tual saturation value. We set equal to the coherent parameter because, in both instances, the integral of the memory for each of the memories divided by the diffusion constant, i.e., over all time exists and equals 1. For the algebraic case ω for exponential, μ for the single delay, and ζ for the algebraic, this is not true and leads to some peculiar behavior. Such respectively. Two curves for each memory depict dynamics in behavior and its analysis given in the present paper will the underdamped regime and one for each memory in the over- no doubt be of importance when parameters controlling damped regime. The inset plot shows that same figure with the non-Markoffian dynamics are strong. Even when they logarithmic axes. are weak, the algebraic nature of the memory will ensure that they must be generally taken into account. while the exponential memory process has a minimum value, V =1exactlywhenb/γ = 1. For these parame- 6.1 Similarities and dissimilarities in the consequences ter values, the algebraic memory has a minimum value of of the three memories V =1/2 located at ν =0. We select values of V>1.19 from the algebraic In all three cases, γφ(t) is convolved with the displace- long-time MSD, equations (33c)and(33d), and solve ment x(t) in the starting Langevin equation. The strength the exponential and single delay memory expressions in of the confinement to the attractive center may be said to equations (30)and(31)toobtain be described by γ and the specific (memory) manner in which the confining is done may be ascribed to φ(t). The ω/b = V ± V 2 − 1 loss of coherence occurs at the rate b in the exponential case, at the rate 1/τ equal to the reciprocal of the delay and time in the single delay case, and at α in the algebraic case. The coherence parameter is ω in the exponential 2 2 2 − 2 √ sin(μ τ )= (2Vμτ) 1 / (2Vμτ) +1 , case, μ = γ/τ for the delay case and ζ = γα in the algebraic case. The three respective memories are found respectively. For the exponential√ memory we choose the − 2 − in equations (5), (19)and(12). negative branch, ω/b = V V 1, for the√ monotonic The basic quantity that determines the behavior of 2 − regime and the positive branch, ω/b = V + V 1, in the system is the Green function λ(t). It is given gener- the oscillatory regime. ally in the Laplace domain by equation (6). In the time Using these values of V , ω/b and μτ we plot in Figure 7 domain it takes the form given in equation (27)forthe the dynamics of the three memories. In the oscillatory exponential memory, equation (26) for the delay mem- regime, all three memories exhibit the apparent satura- ory, and equation (14) for the algebraic case. One of the tion behavior associated with the oscillations of their re- additional results of this paper is an alternate form, equa- spective Green functions. The single delay memory clearly tion (15) that we have derived. We use it along with its saturates the fastest. The exponential and algebraic mem- asymptotic form, equation (18), which is well-known in ories have very similar MSD, with the exponential memory the literature [33]. Both provide considerable computa- initially larger. In the overdamped regime, the algebraic tional convenience. While Figure 1 shows the Green func- memory saturates much slower due to its heavy tail. tion for the algebraic case in the underdamped and overdamped regimes, Figures 2 and 3 display the usefulness of the alternate forms we provide for the computations. 6 Discussion Noteworthy is the fact that the single parameter ν determines in the algebraic case whether oscillatory (posi- The focus of this paper has been the investigation of the tive ν) or monotonically decreasing (negative ν) time vari- dynamics of a Smoluchowski system whose Langevin equa- ation obtains. Whereas for the other two memories, with tion (see, e.g. Eq. (2)) describes the attraction, to a fixed the coherence parameter (ω or μ) held constant, variation Page 12 of 15 Eur. Phys. J. B (2016) 89: 87 of the damping (b or 1/τ respectively) transitions the sys- about the generalization of the Smoluchowski equation. tem from the oscillatory to the overdamped regimes, varia- In our introductory comments in Section 2, we mentioned tion of α with ζ held constant does not do anything similar that, while one could be easily tempted into generaliz- in the algebraic case. As a result of the scaling behavior ing the standard Smoluchowski equation by incorporat- of the power-law dependence, the damping parameter α ing a memory function in its attraction term to describe introduced in the definition (12) of the algebraic mem- the non-Markoffian pull towards the center, such a proce- ory in a manner analogous to the exponential case (5), dure would be incorrect. completely drops out of the picture in the Green function Such a generalization of the time-local Smoluchowski (see Eq. (14)). The coherence parameter merely serves to equation (1) brought about by introducing a memory into scaletimeandν alone determines the oscillatory-decaying the attraction term, would result in transition. This remarkable feature of the algebraic mem- t ory stems from its scale-free nature. ∂P(x, t) ∂ − The time-integral of the Green function from zero to = γ dt φ(t t )xP (x, t ) ∞ ∂t ∂x 0 infinity, i.e., λ(t)dt, is of direct relevance for several ob- 0 ∂2P (x, t) servables. Equation (6) shows that it is given by 1/γφ˜(0). + D . (35) ∂x2 For the exponential and the delay memories, this presents no problems but for the algebraic case one runs into the Although we have seen in Section 2 that it is not possible ˜ ∞ peculiarity that φ(0)maynotexist.Indeed, 0 λ(t)dt to deduce (35) from the Langevin equation with memory vanishes for positive ν but becomes infinite for negative (Eq. (2)), it is interesting to ask what consequences (35) ν provided ν ∈ [−1, 0). One of the direct consequences leads to, since it has the appearance of a natural general- of this feature is the drastic jump from 0 to ∞ observed ization of equation (1) to incorporate a memory process in in the spectral line at zero frequency f noted in Figure 5 the attraction. Through an integration by parts, and using which depicts motional narrowing of the a.c. susceptibility. the physical expectation that P (x, t) vanishes sufficiently For algebraic and delay memories there is also the rapidly at x = ±∞, we obtain equations for the moments regime of unbounded oscillations that occurs for |ν| > 1 of equation (35) by multiplying it by xn and integrating and γτ > π/2, respectively. Because the regime is seldom over x from −∞ to +∞. For the first moment, (n =1), physical, we have shown it only passingly in Figure 4 and we get: only for the delay case, with an effect in the central panel t of Figure 5 where additional peaks in the spectral line dx(t) + γ dt φ(t − t)x(t) =0. (36) result. dt 0

How does this result for the average displacement x(t) 6.2 Non-local attractive term in the Smoluchowski compare with one obtained from the correct general- equation ized Langevin equation (Eq. (2))? Differentiation of equation (9) followed by the use of equation (6) show explicitly Although our focus in this paper is entirely on an anal- that the evolution of the mean displacement as predicted ysis of the Langevin equation (with time non-locality), by the exact (2) is precisely that given by equation (36)! let us recall that there are in the literature, already, While the first moment is given correctly by the inappro- a number of discussions centered on probability den- priate generalization (35), higher moments are not. For sity evolution [29–32]. These approaches aim to con- n>1, the moment evolution from equation (35)is: struct an effective Fokker-Planck equation corresponding to the non-Markoffian Langevin equation (2), in what dxn(t) t has been termed the bona-fide Fokker Planck descrip- + nγ dt φ(t − t)xn(t) tion [30]. Given the memory in the Langevin equation, dt 0 what is required is a Fokker-Planck equation for the con- n! = Dxn−2. (37) ditional probability distribution rather than for the single- (n − 2)! time probability distribution, as would be appropriate in the absence of the memory. Unfortunately, available in To see explicitly that higher moments predicted by the the literature as a Fokker-Planck equation corresponding incorrectly generalized Smoluchowski equation are inac- to equation (2), with Gaussian noise, is only an equa- curate, consider the n = 2 solution of equation (37). We tion for a single-time probability distribution [10,32]. Also, obtain the practical utility of that equation appears not to have t been tested. It is known to be undefined (drift and dif- 2 2 x (t)=λγ→2γ (t)x0 +2D dt λγ→2γ(t ). (38) fusion coefficients blow up) when the mean of the dis- 0 tribution crosses zero [53]. Crossings of this kind occur whenever the Langevin dynamics is oscillatory. It is be- Here, by λγ→2γ is meant what is obtained by replacing γ cause of these uncertainties surrounding Fokker-Planck by 2γ in the expression for the Green function λ(t). Com- treatments of such non-Markoffian cases that we have re- parison to the correct moment (Eq. (14)), shows that the stricted our analysis here to consequences of a Langevin higher moment prediction of the inappropriate general- equation. Let us, nevertheless, return briefly to the issue ization is always inaccurate except for the time-local case Eur. Phys. J. B (2016) 89: 87 Page 13 of 15 when φ(t)=δ(t) leading to an exponential λ(t) and to the The theorem showed that, although frequency-dependent 2 accidental correctness of the relation λγ→2γ (t)=λ (t). quantities require the GME for their correct description, Situations of this kind, wherein lower moments are ac- dc transport coefficients do not: the latter are accurately curately reproduced by an approximate description but described by the Master equation without memory. Al- higher moments are not, are frequently encountered in though the theorem had been presented in detailed deriva- transport theory [54]. In the next subsection we will tions in the original enunciations [55,56], the physical see that, as a consequence of the above, although the understanding and origin of the theorem can be made inappropriate generalization of the Smoluchowski equa- clear quite simply. Let us assume,e.g.inthecontextof tion accurately describes the motional narrowing phe- Ohm’s law, that one has solved for the time-dependent nomenon (given that the latter depends completely on current and thence for a transport coefficient such as the the first moment), it fails to reproduce correctly the size ac conductivity. If the calculation uses a GME with mem- of the particle, time-dependent or steady-state. The size ory φ(t), the expression for the current in the Laplace is determined by the second moment. domain will have inside it φ( ), the Laplace transform of φ(t). The Master equation (which is the GME without memory) would replace φ( )byφ(0) and thus predict 6.3 Apparent violation of the Balescu-Swenson a wrong current in general. theorem However, the Balescu-Swenson theorem pointed out There is a theorem in non-equilibrium statistical mechan- that in calculating a dc transport coefficient such as the dc →∞ ics, named after Balescu [55] and Swenson [56], that states conductivity, equivalently the t limit of the relevant that, while non-Markoffian equations that are the con- observable such as the steady-state current, one would → sequence of microscopic dynamics describe more accu- take the limit 0 as part of a Tauberian argument. One would then get nothing different from the prediction rately the approach to equilibrium or the steady-state than their Markoffian counterparts, their use is unneces- of the memoryless Master equation: in the latter φ( )is sary for describing observables in the steady-state. This replaced by φ(0) from the very beginning. This applies to is so because the latter are reproduced with perfect ac- various steady-state transport coefficients such as the dc curacy by the Markoffian equations. The reasoning be- susceptibility, dc conductivity, viscosity, and ω =0values hind the theorem can be understood by realizing that of lineshapes. the Markoffian approximation involves the replacement by Instances where the Balescu-Swenson theorem does zero of the Laplace variable in the Laplace transforms of not apply have been pointed out earlier [42,63]. Physi- memories; and that asymptotic results also require such cally, the apparent violation has been shown [42] to occur replacement in the implementation of Tauberian theo- when the particle under consideration has a finite lifetime rems. Steady state results are asymptotic results valid at as is the case with an excitation (e.g. a Frenkel exciton): long times. the Balescu-Swenson theorem requires that motion over While the Balescu-Swenson theorem is largely appro- the entire lifetime (or at least over much of it) be played priate and highly useful, its non-judicious and blind ap- out. plication can lead to misleading expectations. The present analysis can provide an interesting example. In order to It was surprising for us to find that the theorem fails understand the context, the usefulness, and the appar- in spite of the fact that we have not assumed a finite life- ent violation of the theorem here, notice that the the- time. The saturation value of the MSD in our system, orem appeared in the work, first of Balescu [55]and in other words the steady-state size of the particle, is an then of Swenson [56], about the time that the gener- asymptotic quantity. Yet we find that its value is depen- alized master equation (GME) made its important ap- dent on the memory function and does not reduce to that pearance. The GME arose in the fundamental work of predicted by the Markoffian approximation (when such is various investigators [57–62]. By dint of the memory func- possible). This is clear from equation (33). tion it possessed, the GME was able to access and de- In order to clarify this further, we have constructed scribe short-time behavior of systems not accessible to Figure 8 to display the time-dependence and the satura- the Master equation. The latter is essentially the GME tion value of the mean square displacement for the sim- with a delta-function memory. An object well-known be- ple case of exponential memory, with γ held constant, for fore the advent of memories, it was the central entity three values of the damping parameter b/γ =0.5, 2.5 of non-equilibrium statistical mechanics and capable of and 12.5. The exact results are shown by dashed lines predicting irreversibility, approach to equilibrium and the and we see that the saturation size increases as b is de- second law of thermodynamics. creased even though γ does not change. This is certainly in Because the GME unravelled numerous features of a conflict with the Balescu-Swenson statement if we assume system additional to those described by the (memory- the saturation value of the particle extent to be a valid less) Master equation, it was of particular importance to steady-state observable in the language of the theorem. know, a priori, what descriptions made it essential to use Figure 8 also shows the interesting result that the incor- the GME in preference to the Master equation and for rectly generalized Smoluchowski equation whose predic- what properties the Master equation was sufficient. The tions are shown in solid line for the corresponding values Balescu-Swenson theorem performed this important task. of the memory parameters do follow the Balescu-Swenson Page 14 of 15 Eur. Phys. J. B (2016) 89: 87

1.5 2 Eq. (30) Eq. (38) Markofﬁan 1.5 Exponential (b/ = 1) Algebraic ( = 0.8) Single Delay ( = 1) 1 1 /2D

2 0.5 x

0.5 f(t,s)/D 0 -0.5

-1 0 0 20 40 60 80 100 120 140 160 180 200 t -1.5 0 5 10 15 20 25 30 Fig. 8. Time-dependence of the extent of the particle (as mea- t-s sured by the value of the MSD) and apparent violation of the Fig. 9. The two-time auto-correlation function plotted as a Balescu-Swenson theorem. The MSD for the exponential mem- function of the difference of the times t and s for all three cases ory is plotted for a constant γ and three values of the damping of the memory function, along with the limit of no memory. parameter b/γ:0.5, 2.5and12.5. The correct results as given We depict f(t, s)/D on the y-axis and t − s on the x-axis, both by our theory, (30), correspond to the dashed lines. Their sat- in units of 1/γ. The sum of the times is held constant at a uration values are seen to increase as b/γ is increased. As all reasonably large value, γ(t+s) = 30 while the difference γ(t−s) cases have the same γ, that increase signifies a violation of is varied over the range [0, 30]. Parameters are chosen so that the Balescu-Swenson theorem. By contrast, all three solid lines the same Markoffian limit is obtained for all three memories, which are predictions of the incorrect generalization, (38), of exponential (dashed line), algebraic (dotted line) and single the Smoluchowski equation (see text) saturate to the same sin- delay (dot-dashed line). We show only the underdamped case. gle value (as given by the time-local equation) even though the corresponding values of b/γ are different (as in the respective dashed line cases). also been given in the context of aging [65,66]. We have provided general expressions for x(t)x(s) and f(t, s)in 2 equation (10)and(11), respectively in terms of the Green theorem. The solid lines all go to the same saturation function λ(t). For instance, for the case of an exponential value of the size that would have been obtained for the memory, we have time-local situation. Needless to say, the original context of the Balescu- D − b (t−s) b ω f(t, s)= e 2 + cos [Ω(t − s)] Swenson theorem did not include systems such as the one ω ω b treated in the present paper. What precise feature of our b2 − ω2 system makes the theorem inapplicable? The simple an- + sin [Ω(t − s)] swer to this question is that, whereas the theorem deals 2Ωω − b (t+s) 4 with situations in which the memory describes the entire be 2 ω process, the memory we consider here refers only to a part − cos [Ω(t − s)] ω Ω2b2 of the process. It appears only in the systematic term of 4Ω2 −ω2 b2 −ω2 the Langevin equation (2). + cos[Ω(t+s)]+ sin [Ω(t+s)] . 4Ω2 2Ωb (39) 6.4 A comment on two-time correlation functions The Markoffian counterpart, obtained by taking the limit 2 Two-time correlation functions, such as x(t)x(s) and b →∞, ω →∞, such that ω /b = γ,is: their antisymmetrized combination f(t, s)whichistermed D in some quarters as their covariance, are of importance f(t, s)= e−γ(t−s) − e−γ(t+s) . (40) in physical situations in which, in addition to the ini- γ tial time, two more times rather than a single one are Neither of these quantities, needless to say, is a func- of importance. As a practical example, consider an elec- − tric current placed initially in a conductor in some man- tion of merely the difference t s. As expected, the non- ner. The current is disappearing as the system relaxes to Markoffian two-time correlation function differs from its Markoffian counterpart at short values of the difference equilibrium. At a certain instant s before equilibrium is − reached, a time-dependent electric field is switched on. At t s but tends to the latter for large values. We depict alatertimet the time-dependent current is measured. The this behavior for the underdamped case graphically in Fig- analysis of such an experimental set-up clearly requires ure 9 for all three cases of the memory by fixing the sum of the two times and varying the difference. Two features are v(t)v(s) , the two-time autocorrelation of the velocity − (instead of x)[64]. Studies of two-time correlations have clear, one at short and one at long t s. As expected, all cases tend to the Markoffian limit at large values of t − s. 2 Care must be taken not to take these predictions of the At short times, on the other hand, they all are different. incorrect generalization of the Smoluchowski equation too se- This is so because there they depict the (dimensionless) riously. There are physical parameter ranges in which the solid mean square displacements which we have seen indeed lines can dip below zero. The negativity is simply a consequence differ, representing the violation of the Balescu-Swenson of the fact that equation (35) is not fully accurate. theorem. Eur. Phys. J. B (2016) 89: 87 Page 15 of 15

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